Identifying Extrema

In Exerc...

Question

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = √(x² − 2x − 3), 3 ≤ x < ∞

Accepted Answer

Hello. In this video, we are going to be considering the function F of X equal to the square root of X2 minus 6 X + 8 in the domain from 4 to infinity. We want to determine its extreme values for this domain and state where they occur. So, before we approach this problem, let's just quickly discuss the domain. Well, we know that the domain of this function is from -4 to positive infinity. The reason why this is important is because we are looking for the extreme values within the domain, and that means that we also have to take the first derivative of this function. So, the function is given to us again as a square root function, but let's go ahead and rewrite this in an exponential form, so that we can easily take the derivative. We will have F of X equal to the quantity X2 minus 6 X + 8 raised to the half power. By using the derivative, or by taking the derivative by using the chain rule, we will get F of X equal to 12 multiplied by the quantity X2 minus 6 X + 8, raised to the negative half power, multiplied by the derivative of the innermost function, which is 2X minus 6. And by combining these terms together, we can simplify the function to be 2 X minus 6 divided by 2 multiplied by the square root of X2 minus 6 X plus 8. Now what we want to do is we want to determine the critical points of this function. Now, in order to determine the critical points, we want to find the values of X that caused the numerator to be zero. So we will take the numerator, 2 X minus 6, set equal to 0 and sulfur X. And by selling for X, we will get the value X is equal to 3. Now, there is one issue with this critical point. This critical point is not insider domain. It is outside the domain since the minimum value of the domain must be -4. So that means that there is not going to be a critical point at X is equal to 3. Let's go ahead and check the value of the function at 4. When we plug in 4 into the original function, we get the square root of 4 squad, which is 16, minus 6 times 4, which is 24, plus 8. And this is going to simplify to be 0. What this means is that this is going to be the minimum value of the function. And now what we can do is we can check the minimum value, or at least the behaviors of the defined part of the domain, from 4 to infinity. We will just go ahead and plug in 5. Into uh into the derivative to determine the behavior. And F of 5 will give us the output of 1.15, which is a positive value. So this tells us that the graph will be increasing from 4 to positive infinity. So, in order to summarize this information, we've identified that we have a local minimum value. At X is equal to 4. And We also determined that there will be no maximum value since this graph is going to be continuing or increasing continuously. And this is going to be the solution to this problem. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = √(x² − 2x − 3), 3 ≤ x < ∞

Key Concepts

Local Extrema

Domain of a Function

Derivative and Critical Points

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